nLab
category of descent data

Idea

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Descent along a cover

This kind of descent is described in the article stack using the bar construction to write a sieve as a colimit of representables.

Let CC be a site, p:D→Cp:D\to C a fibration, x∈C0x\in C_0, UxU_x a cover of xx. hx↪C/xh_x\hookrightarrow C/x the full subcategory of arrows, that factor through some u→xu\rightarrow x of the cover; the corresponding functor Cop→CatC^{op}\rightarrow \Cat is then a sieve. The category of descent data (sometimes shortcut to descent category) of pp along UxU_x is defined to be the functor category [hx,D][h_x,D].